QijinnPm Q coefficients are defined in units of m /C. The M and Q coefficients are equivalent. Conversions between the two coefficients are carried out using the field-polarization relationships Pm=nmEn, and En=X where nmm is the dielectric susceptibility tensor and %mn is the inverse dielectric susceptibility tensor. Electrostriction is not a simple phenomenon but manifests itself as three thermodynamically related effects Sundar and Newnham, 1992]. The first is the well-known variation of strain with polarization, called the irect effect(dx, /dek dE Miu ). The second is the stress(Xy)dependence of the dielectric stiffness xmm,or the reciprocal dielectric susceptibility, called the first converse effect(d/dXy=Mmy). The third effect is the polarization dependence of the piezoelectric voltage coefficient gi, called the second converse effect(dgauldPi Piezoelectricity and Electrostriction Piezoelectricity is a third-rank tensor property found only in acentric materials and is absent in most materials. The noncentrosymmetric point groups generally exhibit piezoelectric effects that are larger than the electro- strictive effects and obscure them. The electrostriction coefficients Miur or Qiu constitute fourth-rank tensors which, like the elastic constants, are found in all insulating materials, regardless of symmetry Electrostriction is the origin of piezoelectricity in ferroelectric materials, in both conventional ceramic rroelectrics such as BaTiO, as well as in organic polymer ferroelectrics such as PVDF copolymers[ Furukawa and Seo, 1990]. In a ferroelectric material, that exhibits both spontaneous and induced polarizations, P: and Pi, the strains arising from spontaneous polarizations, piezoelectricity, and electrostriction may be formulated xi=Qik PPi+ 2QikPP'+ Qir PpP (50.4) In the paraelectric state, we may express the strain as x=Qm Pp P, so that dx /dpi=8k=2QjuPr Converting to the commonly used d coefficients dijk=mk gim=2x mk Qijmn Pn This origin of piezoelectricity in electrostriction provides us an avenue into nonlinearity. In this case, it is the ability to tune the piezoelectric coefficient and the dielectric behavior of a transducer. The piezoelectric coefficient varies with the polarization induced in the material, and may be controlled by an applied electric field. The electrostrictive element may be tuned from an inactive to a highly active state. The electrical impedance of the element may be tuned by exploiting the dependence of permittivity on the biasing field for these materials, and the saturation of polarization under high fields [ Newnham, 1990] Electrostriction and Compliance Matrices The fourth-rank electrostriction tensor is similar to the elastic compliance tensor, but is not identical. Com- pliance is a more symmetric fourth-rank tensor than is electrostriction. For compliance, in the most general case, (50.6) but for electrostriction: Mil=Mial=Milk=Mi* Mkli-- MIki=Mali= Mikil c 2000 by CRC Press LLC© 2000 by CRC Press LLC xij = QijmnPm Pn (50.2) Q coefficients are defined in units of m4 /C2 . The M and Q coefficients are equivalent. Conversions between the two coefficients are carried out using the field-polarization relationships: Pm = hmnEn , and En = cmn Pm (50.3) where hmn is the dielectric susceptibility tensor and cmn is the inverse dielectric susceptibility tensor. Electrostriction is not a simple phenomenon but manifests itself as three thermodynamically related effects [Sundar and Newnham, 1992]. The first is the well-known variation of strain with polarization, called the direct effect (d2 xij/dEk dEl= Mijkl). The second is the stress (Xkl) dependence of the dielectric stiffness cmn, or the reciprocal dielectric susceptibility, called the first converse effect (dcmn/dXkl = Mmnkl). The third effect is the polarization dependence of the piezoelectric voltage coefficient gjkl , called the second converse effect (dgjkl/dPi = cmkcnlMijmn). Piezoelectricity and Electrostriction Piezoelectricity is a third-rank tensor property found only in acentric materials and is absent in most materials. The noncentrosymmetric point groups generally exhibit piezoelectric effects that are larger than the electro￾strictive effects and obscure them. The electrostriction coefficients Mijkl or Qijkl constitute fourth-rank tensors which, like the elastic constants, are found in all insulating materials, regardless of symmetry. Electrostriction is the origin of piezoelectricity in ferroelectric materials, in both conventional ceramic ferroelectrics such as BaTiO3 as well as in organic polymer ferroelectrics such as PVDF copolymers [Furukawa and Seo, 1990]. In a ferroelectric material, that exhibits both spontaneous and induced polarizations, P s i and P¢ i , the strains arising from spontaneous polarizations, piezoelectricity, and electrostriction may be formulated as (50.4) In the paraelectric state, we may express the strain as xij = QijklPkPl, so that dxij/dPk = gijk = 2QijklPl. Converting to the commonly used dijk coefficients, dijk = cmk gijm = 2cmkQijmnPn (50.5) This origin of piezoelectricity in electrostriction provides us an avenue into nonlinearity. In this case, it is the ability to tune the piezoelectric coefficient and the dielectric behavior of a transducer. The piezoelectric coefficient varies with the polarization induced in the material, and may be controlled by an applied electric field. The electrostrictive element may be tuned from an inactive to a highly active state. The electrical impedance of the element may be tuned by exploiting the dependence of permittivity on the biasing field for these materials, and the saturation of polarization under high fields [Newnham, 1990]. Electrostriction and Compliance Matrices The fourth-rank electrostriction tensor is similar to the elastic compliance tensor, but is not identical. Com￾pliance is a more symmetric fourth-rank tensor than is electrostriction. For compliance, in the most general case, sijkl = sjikl = sijlk = sjilk = sklij = slkij = sklji = slkij (50.6) but for electrostriction: Mijkl = Mjikl = Mijlk = Mjilk ¹ Mklij = Mlkij = Mklji = Mlkij (50.7) x Q P P Q P P Q P P ij ijkl k s l s ijkl k s l ijkl k l = + 2 ¢ + ¢ ¢